matrix – Manipulation list: {{{a, b}, {c, d}}, {{e, f}, {gh}}} in {{a, b}, {c, d}, {e, f}, {g, h}}

Hello, I would like to transform the matrix:

{{{184.586, 1.02758 * 10 ^ 8}, {139.94, 4.07249 * 10 ^ 7}, {117.72,
2.26123 * 10 ^ 7}, {109.528, 1.51412 * 10 ^ 7}, {95.7636,
8.68593 * 10 ^ 6}, {78.125, 5.16118 * 10 ^ 6}, {66.6777,
3.1014 * 10 ^ 6}}, {{183.505, 1.01147 * 10 ^ 8}, {138.25,
4.02021 * 10 ^ 7}, {115.814, 2.24641 * 10 ^ 7}, {108.22,
1.50985 * 10 ^ 7}, {91.3082, 9.28241 * 10 ^ 6}, {78.1087,
6.03119 * 10 ^ 6}, {65.9855, 3.56718 * 10 ^ 6}}}

with the same transformation that

{{{a, b}, {c, d}}, {{e, f}, {gh}}} 

inside

{{a, b}, {c, d}, {e, f}, {g, h}}

How can I do please?

gh

hg

rt.representation theory – Expressing $ sum_ {g in [G/H]} ge_Hg ^ {- 1} in Z ( mathbb {C}[G]) $ in terms of primitive central idempotentes?

Suppose $ G $ it's a finite group, and $ H $ a subgroup For an irreducible character. $ chi $ of $ G $, there is a central idempotent in group algebra. $ mathbb {C}[G]$:
$$
e_ chi = frac { chi (1)} {| G |} sum_ {g in G} chi (g ^ {- 1}) g.
$$

I write $ e_H: = e_ {1_H} = frac {1} {| H |} sum_ {h in H} h $ for the idempotent in $ mathbb {C}[H]$ corresponding to the trivial character of $ H $. Yes $[G/H]$ Denotes a complete set of left coset representatives from $ H $ in $ G $, then by construction the element
$$
sum_ {g in [G/H]} ge_Hg ^ {- 1}
$$

it is central in $ mathbb {C}[G]$. Is there any way to explicitly extract which characters have their corresponding central idempotent in this linear combination and / or its multiplicity? Just looking at Mackey's formula and since then $ ge_Hg ^ {- 1} = e_ {gHg ^ {- 1}} $, my guess is that they can be characters that are constituents of $ operatorname {Ind} ^ G_H (1_H) $, or something similar, but I'm not sure.

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